Theorem bnj1326

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1326.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1326.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1326.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1326.4	⊢ 𝐷 = ( dom 𝑔 ∩ dom ℎ )
Assertion	bnj1326	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) → ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1326.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1326.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1326.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1326.4	⊢ 𝐷 = ( dom 𝑔 ∩ dom ℎ )
5		eleq1w	⊢ ( 𝑞 = ℎ → ( 𝑞 ∈ 𝐶 ↔ ℎ ∈ 𝐶 ) )
6	5	3anbi3d	⊢ ( 𝑞 = ℎ → ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) ) )
7		dmeq	⊢ ( 𝑞 = ℎ → dom 𝑞 = dom ℎ )
8	7	ineq2d	⊢ ( 𝑞 = ℎ → ( dom 𝑔 ∩ dom 𝑞 ) = ( dom 𝑔 ∩ dom ℎ ) )
9	8	reseq2d	⊢ ( 𝑞 = ℎ → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
10	4	reseq2i	⊢ ( 𝑔 ↾ 𝐷 ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom ℎ ) )
11	9 10	eqtr4di	⊢ ( 𝑞 = ℎ → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ 𝐷 ) )
12	8	reseq2d	⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
13		reseq1	⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom ℎ ) ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
14	12 13	eqtrd	⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) ) )
15	4	reseq2i	⊢ ( ℎ ↾ 𝐷 ) = ( ℎ ↾ ( dom 𝑔 ∩ dom ℎ ) )
16	14 15	eqtr4di	⊢ ( 𝑞 = ℎ → ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( ℎ ↾ 𝐷 ) )
17	11 16	eqeq12d	⊢ ( 𝑞 = ℎ → ( ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ↔ ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) ) )
18	6 17	imbi12d	⊢ ( 𝑞 = ℎ → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) → ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) ) ) )
19		eleq1w	⊢ ( 𝑝 = 𝑔 → ( 𝑝 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶 ) )
20	19	3anbi2d	⊢ ( 𝑝 = 𝑔 → ( ( 𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) ) )
21		dmeq	⊢ ( 𝑝 = 𝑔 → dom 𝑝 = dom 𝑔 )
22	21	ineq1d	⊢ ( 𝑝 = 𝑔 → ( dom 𝑝 ∩ dom 𝑞 ) = ( dom 𝑔 ∩ dom 𝑞 ) )
23	22	reseq2d	⊢ ( 𝑝 = 𝑔 → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑝 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
24		reseq1	⊢ ( 𝑝 = 𝑔 → ( 𝑝 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
25	23 24	eqtrd	⊢ ( 𝑝 = 𝑔 → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
26	22	reseq2d	⊢ ( 𝑝 = 𝑔 → ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
27	25 26	eqeq12d	⊢ ( 𝑝 = 𝑔 → ( ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) ↔ ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ) )
28	20 27	imbi12d	⊢ ( 𝑝 = 𝑔 → ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) ) ) )
29		eqid	⊢ ( dom 𝑝 ∩ dom 𝑞 ) = ( dom 𝑝 ∩ dom 𝑞 )
30	1 2 3 29	bnj1311	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑝 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑝 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑝 ∩ dom 𝑞 ) ) )
31	28 30	chvarvv	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ 𝑞 ∈ 𝐶 ) → ( 𝑔 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) = ( 𝑞 ↾ ( dom 𝑔 ∩ dom 𝑞 ) ) )
32	18 31	chvarvv	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑔 ∈ 𝐶 ∧ ℎ ∈ 𝐶 ) → ( 𝑔 ↾ 𝐷 ) = ( ℎ ↾ 𝐷 ) )